Sup-norms of eigenfunctions on arithmetic ellipsoids - TAN | EPFL

SUP-NORMS OF EIGENFUNCTIONS ON ARITHMETIC ELLIPSOIDS VALENTIN BLOMER AND PHILIPPE MICHEL MARCH 6, 2011 Abstract. Let B be a positive quaternion algebra, and let O ⊆ B be an Eichler order. There is associated in a natural way a variety X = X(O) whose connected components are indexed by the ideal classes of O and are isomorphic to spheres. This variety is naturally equipped with a Laplace operator and a large family of Hecke operators. For a joint eigenfunction ϕ of the Hecke algebra and of the Laplace operator with eigenvalue λ, the hybrid sup norm bound kϕk∞  (tV )−δ t1/2 kϕk2 for any δ < 1/60 is shown, where t = (1 + λ)1/2 and V = vol(X(O)).

1. Introduction 1.1. General setup. Let X be a compact Riemanian manifold and let ∆ denote the Laplace operator. A central issue is the study of the behaviour of Laplace eigenfunctions ∆ϕ + λϕ = 0, as λ → +∞; in particular, a classical problem is to estimate the L∞ norm kϕk∞ . We refer to [Sar] for a very general and enlightening description of the latter problem along with very precise conjectures about what to expect. For instance, when X is 2-dimensional, the following standard bound holds: (1.1)

kϕk∞ X (1 + λ)1/4 kϕk2 .

This bound is essentially sharp, as is seen by considering X = S 2 and ϕ a zonal spherical function (the restriction to S 2 of a harmonic homogeneous polynomial in R3 which is invariant by the group of rotations fixing a given point on S 2 ). On the other hand, on hyperbolic surfaces stronger bounds are expected as an effect of negative curvature. These expectations are supported by the groundbreaking work of Iwaniec and Sarnak [IS95] who proved the following bound for X a Riemann surface coming from an indefinite quaternion algebra over Q (a Shimura curve) and ϕ a Hecke-Laplace eigenform: (1.2)

1

1

kϕk∞ X (1 + λ) 4 − 24 +ε kϕk2 .

1 When X is compact, the above bound is conjectured to hold with 41 − 24 replaced by 0. The discrepancy between the positive curvature case (say the sphere) and the hyperbolic case is closely related to the fact that in the former, Laplace eigenvalues have high multiplicities while in the latter, the multiplicities are expected to be essentially bounded. Very little is known about these multiplicities in the hyperbolic case (even for arithmetic surfaces), and this explains why Iwaniec and Sarnak considered Hecke eigenforms for which multiplicity one theorems hold1. In the present paper, we reconsider and extend the previous reasoning to the 2-sphere which may be realized as a connected component of a locally homogeneous space of arithmetical type associated with a definite quaternion algebra defined over Q. It is therefore equipped with a large commutative ring of “Hecke operators” commuting with ∆. Considering Laplace eigenfunctions which are also

2000 Mathematics Subject Classification. 11R52, 11F72, 11E20, 58J50. Key words and phrases. definite quaternion algebras, trace formula, sup-norm, hybrid bounds, norm forms, spherical harmonics. V. B. is supported by a Volkswagen Lichtenberg grant. Ph. M. is partially supported by the ERC advanced research grant n. 228304 and the SNF grant 200021-12529. 1of course if multiplicities are very small, as expected, restricting to Hecke eigenforms does not reduce the generality 1

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eigenfunctions of the Hecke operators eliminates the issue of high multiplicities, and we can obtain a bound analogous to (1.2) (with 1/24 replaced by 1/48, cf. (5.13) below). 1.2. Hybrid bounds. We should point out, however, that our main interest here is somewhat different from the setup in [IS95] as we are interested in how (1.1) depends on X. This question was investigated in great generality in [Don01] where it is shown that the dependency involves the sectional curvature and the injectivity radius. More precise results are available for locally symmetric spaces Γ\G/K. For suitable families of hyperbolic surfaces XΓ = Γ\H (including suitable coverings of a fixed surface [Rho93, JK04] or arithmetic surfaces), the bound (1.1) takes the more precise form (1.3)

kϕk∞  vol(X)o(1) (1 + λ)1/4 kϕk2 ,

where the metric on XΓ is the one descending from a fixed hyperbolic metric on H. It is natural to try to improve on (1.3) simultaneously in the eigenvalue and the “volume” aspect. Such bounds are called “hybrid”. The first example of such a hybrid bound was obtained in the work of Holowinsky and the first named author [BH10]: as XΓ varies over the sequence of (non-compact but finite volume) modular curves of level squarefree level N , X0 (N ) = Γ0 (N )\H∗ → X0 (1) = SL2 (Z)\H∗ ,

N squarefree,

one has, for any Hecke-Maass cuspform ϕ on X0 (N ), (1.4)

kϕk∞  (1 + λ)1/2 vol(X0 (N ))


−δ

(1 + λ)1/4 kϕk2

where δ > 0 is an absolute constant (let us recall that vol(X0 (N )) = N 1+o(1) ). The best possible bound in this situation would be δ = 1/2 + o(1) which among other things would imply the Lindel¨of hypothesis for the Hecke L-function L(ϕ, s) (see §6 for a related discussion). In the present paper, we obtain a hybrid analog of (1.4) for 2-spheres. As we explain in Section 4 (see in particular (4.1)), to a definite quaternion algebra B defined over Q and an order O ⊂ B(Q), there is associated a finite disjoint union of (finite quotients of) spheres S 2 : G X := X(O) = X[I] , X[I] := Γ[I] \S 2 , [I]∈Cl(O)

Γ[I] < SO3 (R) some finite subgroup of order ≤ 12. The quotients X[I] are the components of X and the (finite, but possibly large) set indexing the components is the set of classes of O-ideals. The variety X is equipped with a “natural” Riemannian metric and a corresponding volume form: the metric is obtained by choosing once and for all a left SO3 (R)-invariant Riemannian metric on S 2 and by putting on each component X[I] the induced metric multiplied by 1/|Γ[I] |; in this way vol(X[I] )  1, and the volume of X has roughly the size of the class group: vol(X(O))  |Cl(O)| = disc(O)1/2+o(1) , cf. (2.4). As a Riemannian manifold, X is equipped with a Laplace operator ∆. Moreover, being arithmetically defined, X is also endowed with a commutative algebra of Hecke correspondences T which is generated by Hecke correspondences (Tp )p , indexed by the primes p coprime with disc(O). Each correspondence Tp is of degree p + 1 and yields a selfadjoint Hecke operator (w.r.t. the measure derived from the Riemannian metric) commuting with ∆. Our main result is the following: Theorem 1. Let X(O) be as above and suppose that O is an Eichler order. Let ϕ be a Laplace eigenform ϕ with eigenvalue λ which is also an eigenform of the Hecke algebra T. One has
−δ 1 kϕk∞  (1 + λ)1/2 vol(X(O)) (1 + λ) 4 kϕk2 for some absolute constant δ > 0. Any value δ < 1/60 is admissible.

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Remark 1.1. The result is non-trivial already for the trivial eigenvalue λ = 0. In this case ϕ is constant on each of the components X[I] . The space of locally constant functions has a very deep number theoretic meaning. For instance, if B is the definite quaternion algebra ramified at a single prime q and O is a maximal order, this space is identified with the set of function on the set of isomorphy classes of supersingular elliptic curves in characteristic q (see [Gro87, Section 2]). The space of such functions has large dimension, equal to |Cl(O)| ≈ vol(X(O)). Again we see that (1.3) is in general sharp in the volume aspect: pick the function which is 1 on a given component and 0 on the others (a similar reasoning involving harmonic polynomials instead of constant functions shows that (1.3) is sharp in both aspects). However, the restriction to Hecke eigenfunction resolves this multiplicity issue, and the meaning of kϕk∞  vol(X(O))−δ kϕk2 is an equidistribution statement for Hecke correspondences: a locally constant function which is moreover a Hecke-eigenfunction cannot accumulate too much of its mass on a single component. Remark 1.2. There are a priori two main “directions” in which X = X(O) could vary: either by varying the order O inside B (the main numerical parameter measuring the variation of the Eichler order O is its level N which is a positive integer coprime with the discriminant of B) or by varying B (which is equivalent to letting disc(B) → ∞). Our result is uniform in both of these directions of variation. In [BH10] only the O-direction was considered (the corresponding quaternion algebra in that case is the algebra B = M2 (Q) of 2 × 2 matrices). However, the results of [BH10] remain valid also when B varies amongst the indefinite quaternion algebras, see [Tem] for a nice extension. Remark 1.3. The numerical value of δ follows from an inspection of (5.12) and (5.13) below. Our aim was to show the existence of some δ > 0 with relatively little technology; with more involved estimations, the exponent 1/60 could be improved. Remark 1.4. Another motivation to study sup-norms of eigenfunctions comes from a Waldspurgertype formula due to Gross (see [Gro87, Hat90]) that relates central values of certain Rankin-Selberg L-functions to averages of automorphic forms over CM-points. In this way, the bound of Theorem 1 in the λ-aspect translates into a subconvex bound for certain Rankin-Selberg L-functions L(π1 ⊗π2 , 1/2) in terms of the archimedean parameter of π1 . Although there are by now numerically stronger and more general subconvexity results, the method is new and very different from the usual approaches and based only on the arithmetic of quadratic forms. See Section 6 for a more detailed discussion. 1.3. Principle of proof. Our proof proceeds roughly as follows. We consider an amplified second moment which we transform by a pre-trace formula into a sum over a sort of automorphic kernel, see (5.3). This starting point is similar in most investigations of sup-norms of eigenfunction on arithmetically defined manifolds, see e.g. [IS95, BH10, Mil, Tem]. In all cases one encounters eventually an interesting diophantine problem whose solution is at the heart of the problem. Here the analysis diverges in all known cases, and depends in a non-trivial way on the underlying manifold. In our situation, the sum over the automorphic kernel can be expressed in terms of weighted sums of representation numbers of integers by definite quaternionic norm forms, cf. for instance (5.5) and the subsequent remark which interprets this expression as an average over Fourier coefficients of certain theta-series. These sums can be bounded by the classical reduction theory of quadratic forms and methods from diophantine approximation, but the analysis is quite subtle. To get a non-trivial sup-norm bound in the λ-aspect, one has to estimate the number of integral solutions to Q(x) = ` close to two given orthogonal hyperplanes for certain quaternary quadratic forms Q. To get a non-trivial sup-norm bound in the volume aspect, one has to estimate the number of solutions to Q(x) = ` for certain quaternary forms Q of very large discriminant (compared with `). These two bounds are given in Lemmas 1 and 2 that appear to be new and may also be useful in other contexts. These two results can be combined to yield a simultaneous bound in both aspects.

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The article is organized as follows: Section 2 compiles and recalls basic facts on quaternion algebras. Section 3 is the diophantine heart of the paper and contains bounds for representation numbers of quadratic forms that are needed later. Section 4 realizes the manifold X as an adelic quotient of a quaternion algebra and introduces the relevant operators on this space. In section 5 we construct a suitable test function and a suitable amplifier for the pre-trace formula and estimate the resulting expression. Section 6 provides a link to the subconvexity problem. 1.4. Possible extensions. The results of this paper can be generalized in various directions. Firstly, a Laplace eigenfunction on S 2 may be viewed as a function on the group SO3 (R), left-invariant under some suitably embedded SO2 (R) which is an eigenform of the Casimir operator; the method presented here extends readily and provides non-trivial bounds for the more general Hecke-Casimir eigenfunctions. Less evidently perhaps, these methods extend to Hecke-Laplace (and Hecke-Casimir) eigenfunctions associated to totally definite quaternion algebras defined over a fixed totally real number field (the associated symmetric space is then a union of quotients of products of spheres.) A nice consequence of this extension to more general number fields is that such bounds provide nontrivial bounds for Hecke-Laplace (more generally Hecke-Casimir) eigenfunctions on certain unions of 3-dimensional ellipsoids (more generally orthogonal groups) associated to definite quaternary quadratic forms over Q. For instance, the result of the present paper essentially provides such bounds when the discriminant of the quaternary form is a square, while the case of a non-square discriminants follows from the above mentioned generalization to the quadratic extension generated by the square root of the discriminant. Such extensions will be discussed in a subsequent paper. Acknowledgement. The first author would like to acknowledge the hospitality and the excellent working conditions at the EPFL. The authors would also like to thank the referee for a very careful reading of the manuscript. 2. Arithmetic in quaternion orders 2.1. Quaternion algebras. We recall basic terminology and facts of quaternion algebras, see   some a,b × e.g. [Vig80]. For a, b ∈ Q , let B = Q be the corresponding quaternion algebra over Q, i.e. B(Q) = Q + Qi + Qj + Qij with i2 = a, j 2 = b, ij + ji = 0. Its center Z(B) is the algebra of scalars Q. If γ = x1 + x2 i + x3 j + x4 ij ∈ B, let γ¯ := x1 − x2 i − x3 j − x4 ij be the canonical involution and let tr and nr be the reduced norm and trace 1 tr(γ) = γ + γ¯ = 2x1 , nr(γ) = γ¯ γ = tr(γγ) = x21 − ax22 − bx23 + abx24 . 2 A place v is called ramified if Bv := B ⊗Q Qv is a division algebra, and non-ramified otherwise; in the former case, Bv is the unique (up to isomorphism) quaternion division algebra over Qv ; in the latter Bv ∼ = Mat(2, Qv ) in which case the reduced norm and reduced trace are given by the usual determinant and trace and for matrices. We recall that a quaternion algebra is ramified at an even finite number of places and that the finite ramified places divide 2ab. For the rest of this paper we assume that a, b < 0 so that B is ramified at ∞ (i.e. B(R) is the algebra of real Hamilton quaternions). The reduced discriminant DB of B is the product of the finite ramified primes. 2.2. Lattices, orders. A lattice or ideal I ⊂ B(Q) is a Z-module of maximal rank 4. The product of two lattices I1 , I2 is given by I1 I2 := {γ1 γ2 | γ1 ∈ I1 , γ2 ∈ I2 }, and the inverse is given by I−1 := {γ ∈ B | IγI ⊆ I}. An order O is a subring of B(Q) which is also lattice, in particular its elements are integral (that is, tr(γ), nr(γ) ∈ Z for γ ∈ O). It follows that an element γ ∈ O is invertible if and only if nr(γ) = 1, and one can show |O × | ≤ 24 (see [Vig80, p. 145]). The left (resp. right) order Ol (I) (resp. Or (I)) of a lattice I is the set Ol (I) = {γ ∈ B(Q), γI ⊂ I}

(resp. Or (I) = {γ ∈ B(Q), Iγ ⊂ I}).

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The left (resp. right) order of a lattice is an order. Given an order O, a left O-ideal is defined as a lattice I such that Ol (I) = O (right O-ideals are defined in the obvious way). Two right O-ideals I, I0 are called equivalent if there exists γ ∈ B× (Q) such that I0 = γI. The set of such equivalence classes is denoted Cl(O); this set is finite and its cardinality |Cl(O)| is the class number of O. 2.3. Localization. Given a lattice I, we denote by Ip = I ⊗Z Zp the closure of I inside Bp . The choice of a maximal order Omax ⊂ B(Q) and of a Z-basis of Omax determines, for every unramified p (i.e., not dividing DB ), isomorphisms (2.1)

Omax,p ' M2 (Zp ),

hence of Qp -algebras Bp ' M2 (Qp ).

We fix such a choice and in the sequel identify freely elements of Bp (for p unramified) with matrix elements. On the other hand if p is ramified, then Omax,p = {γp ∈ Bp , nr(γp ) ∈ Zp } is the unique maximal order of Bp . Two right O-ideals I, I0 are called everywhere locally equivalent if for every prime p there is 0 γp ∈ B × p such that Ip = γp Ip . This equivalence relation is coarser than the previous one, and having everywhere only one local equivalence class for a given order O is equivalent to saying that every right O-ideal is everywhere locally principal. 2.4. Discriminant and reduced discriminant. One has 2nr(γ) = tr(γγ), so the trace defines a non-degenerate bilinear form 1 (γ, γ 0 ) = tr(γγ 0 ). 2 The discriminant of an order O is by definition2 disc(O) = det(tr(γi γ j )i,j≤4 ) for {γ1 , . . . , γ4 } a Z-basis of O. This does not depend on the choice of the basis. The reduced norm nr(I) ∈ Q of a lattice I is the positive generator of the fractional Z-ideal generated by all elements nr(γ) with γ ∈ I. The reduced norm is multiplicative on ideals [Vig80, p. 24]. The dual of a lattice I is the lattice I∗ = {γ ∈ B(Q), tr(γI) ⊂ Z}. The different (or complement in Eichler’s terminology) of an order O is the dual O ∗ of O. This is an O-module and one defines the reduced discriminant disc∗ (O) of O to be (2.2)

disc∗ (O) := nr((O ∗ )−1 ) = nr(O ∗ )−1

(by multiplicativity of the norm). If O is a maximal order, its reduced discriminant equals the reduced discriminant DB of B defined at the beginning of this section [Vig80, II.4.7]. One has the following relation between discriminant and reduced discriminant [Vig80, I.4.7]3 (2.3)

disc(O) = disc∗ (O)2 .

All these invariants admit of course local counterparts and the global ones are obtained as products of the local ones. 2Often the discriminant of O is defined as det(tr(γ γ ) i j i,j≤4 ) which differs from our definition by a sign. 3Strictly speaking, the proof in [Vig80] is only carried out in the class number one case and left as an exercise in

the general case. In the present paper we consider Eichler orders (see below) and this special case is already enough, since Eichler orders have locally class number one, and both sides of (2.3) are the products of their local components. If the reduced discriminant is squarefree, (2.3) is easy to see anyway, e.g. [Eic55, p. 131, first paragraph].

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2.5. Eichler orders. An Eichler order is by definition the intersection of two maximal orders. To an Eichler order there is associated an integer4 N coprime to DB such that for every p coprime with Z Z DB , Op is conjugate to the order {( N Zpp Zpp )} of M2 (Qp ). We may and will assume that the choice of Omax is such that Op corresponds precisely to {( N Zpp Zpp )} under the identification (2.1). Note also that since Op is the unique maximal order of Bp at a ramified prime p, Eichler orders associated with the same N are locally conjugate (and conversely). Finally, since right (resp. left) ideals of an Eichler order O are locally principal (that is, for every p there is γp ∈ B× (Qp ) such that Ip = γp Op , cf. [Vig80, Section 2]), the left order Ol of a right O-ideal is everywhere locally conjugate to O (that is, (Ol )p = γp Op γp−1 ). In particular Ol is conjugate to O by an element of B× (Q). For an Eichler order O, the discriminant and the reduced discriminant have the following explicit expressions [Vig80, p. 85] Z

disc∗ (O) = DB N,

Z

disc(O) = (DB N )2 ,

and the class number equals [Vi, p. 143]   −1 Y  1 1 Y 1/2+o(1) = disc(O) . 1− (2.4) |Cl(O)|  DB N 1− p p p|N

p|DB

3. Representation numbers of quadratic forms In this section we provide the necessary tools to treat the diophantine problems mentioned at the end of the introduction. We recall some facts about positive definite quadratic forms. Let 1 X Q(x) = aij xi xj , aij = aji ∈ Z, ajj ∈ 2Z, 2 1≤i,j,≤n

be a positive definite integral quadratic form in n variables. Let A = (aij )1≤i,j≤n be the integral, even (i.e. the diagonal elements are even), symmetric n×n-matrix associated to Q. The determinant of Q is just ∆ = det A, and the level of Q is the smallest integer N such that N A−1 is an integral even matrix. Both the determinant and the level are the products of the their local components (that is, the determinant and the level of the quadratic lattices (Znp , Q) for p varying over the primes), and hence are same for two quadratic forms that are everywhere locally equivalent. 3.1. Reduction theory. A form Q is called Minkowski-reduced if for all j = 1, . . . , n and for all x ∈ Z4 such that (e1 , . . . ej , x) can be extended to an integral basis of Zn , one has Q(x) ≥ Q(ej ), cf. [Cas78, ch. 12]. Every Q is Z-equivalent to a Minkowski-reduced form ([Cas78, Theorem 12.1.1]). It is not hard to see [Cas78, Lemma 12.1.1] that a Minkowski-reduced form satisfies 0 < a11 ≤ a22 ≤ . . . ≤ ann ,

(3.1)

|2aji | ≤ ajj for 1 ≤ j < i ≤ n.

We can write a Minkowski-reduced form Q as X Q(x) = aij xi xj = h1 (x1 + c12 x2 + . . . + c1n xn )2 + h2 (x2 + c23 x3 + . . . + c2n xn )2 + . . . + hn x2n . 1≤i,j≤n 4In [Vig80] and many other places this integer is called the “level” of the Eichler order. Eichler [Eic55, §2] uses the word level for what we called the reduced discriminant. To avoid confusion, we will not use the word level for an order, but only for the associated norm form, and we will use instead the terminology “an Eichler order associated to the integer N ”.

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Then it is known [Cas78, Theorem 12.3.1] that ajj  hj where the implied constants in the upper and lower bound depend only on n. Let us define Hj := min hi  ajj .

(3.2)

i≥j

We see inductively that Q(x) ≥ Hj unless xj = xj+1 = . . . = xn = 0.

(3.3) We denote generally by

0 < λ1 ≤ . . . ≤ λn

(3.4)

the eigenvalues of A. By (3.1) and (3.2) we have λn  ann  Hn  det A.

(3.5)

˜ det A  1 where A˜ is an (n − 1) × (n − 1) By Cramer’s rule, the entries of A−1 are given by det A/ minor of A. Hence λ1  1.

(3.6)

Let µn denote the n-dimensional Lebesgue measure. Then µn ({x ∈ Rn | Q(x) ≤ y}) n

(3.7)

y n/2 . ∆1/2

Given y > 0, we denote by µn−1 {x ∈ Rn | Q(x) = y} the measure on the ellipsoid deduced from the restriction of the euclidean metric on Rn ; one has 1/2

(3.8)

µn−1 ({x ∈ Rn | Q(x) = y}) n

y (n−1)/2 y (n−1)/2 Hn y (n−1)/2 = n , 1/2 1/2 (λ1 · . . . · λn−1 ) (∆/λn ) ∆1/2

since the axes of the ellipsoid Q(x) = y have lengths (y/λj )1/2 . We use the notation rQ (`) to denote the number of integral representations of ` by Q. 3.2. Diophantine lemmas. This section contains the two diophantine results that will eventually yield nontrivial bounds in the volume aspect and the λ-aspect in Theorem 1. Lemma 1. Let Q be a positive-definite integral quaternary quadratic form of determinant ∆ and level N . For y ≥ 1 let A ⊆ N ∩ [1, y] be a subset of integers bounded by y. Then X

rQ (`) ε

`∈A

y2 y 3/2 + + y ε |A| ∆1/2 (∆/N )1/2

for any ε > 0, the implied constant depending on ε alone. Proof. We can assume that Q is Minkowski-reduced. Let A˜ = (aij )1≤i,j,≤3 be the upper left ˜ its determinant. Clearly A˜ is again a Minkowski-reduced 3 × 3-submatrix of A, and denote by ∆ ˜ matrix. The entry in the lower right corner of A−1 is by Cramer’s rule ∆/∆; by definition of the ˜ level, N ∆/∆ must be integral, hence ˜ ≥ ∆. ∆ N

(3.9)

We distinguish three cases. If y ≥ H4 , then by (3.7) and (3.8) we have by the Lipschitz principle 1/2

X `∈A

rQ (`) ≤

X `≤y

rQ (`) 

y2 y 3/2 H4 + 1/2 ∆ ∆1/2



y2 . ∆1/2

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VALENTIN BLOMER AND PHILIPPE MICHEL MARCH 6, 2011

If H3 ≤ y < H4 , then by (3.3), all ` are represented by the ternary form corresponding to the matrix ˜ and hence by (3.9) and the same reasoning we find A, 1/2

X

rQ (`) 

`∈A

y 3/2 yH3 y 3/2 y 3/2 +   . ˜ 1/2 ˜ 1/2 ˜ 1/2 (∆/N )1/2 ∆ ∆ ∆

Finally if y < H3 , then again by (3.3) all ` are represented by a binary form, so that rQ (`)  y ε . This completes the proof of the lemma.  For the next lemma we introduce the following notation: For a positive-definite integral quaternary quadratic form Q given by a matrix A and some parameter ξ ≥ 0, we write rQ (`, ξ) := sup |{x ∈ Z4 | Q(x) = `,

|vT1 Ax|2 + |vT2 Ax|2 ≤ ξ}|

where the supremum is taken over all pairs v1 , v2 of orthonormal vectors in the quadratic space (R4 , Q), that is (3.10)

Q(v1 ) = Q(v2 ) = 1,

vT1 Av2 = 0.

Lemma 2. Let Q be a positive-definite integral quaternary quadratic form of determinant ∆. Let ` ∈ N and ξ > 0. Then rQ (`, ξ) ε (1 + ξ∆)(1 + ∆2 ξ + (∆2 `7 ξ)1/8 )(`∆(ξ + 1/ξ))ε , with an implied constant depending only on ε. Moreover, rQ (`, ξ) ≤ rQ (`) ε `1+ε . Remark. This lemma bounds the number of representations of ` by Q whose projection to a given 2-dimensional plane is small as measured by the parameter ξ. Qualitatively, that lemma says that if ξ is a large enough negative power of ∆`, then the number of such representation is essentially bounded. Such qualititive statement is already sufficient to deduce a non-trivial bound for kϕk∞ in the λ-aspect. Proof. For small ξ, we need to count lattice points in some slightly thickened S 1 inside S 3 . In other words, we have a slightly perturbed binary problem which explains why the number of representations should be almost bounded. To make this idea precise, we denote the eigenvalues of A as in (3.4) and write A = B T B for some (unique) positive symmetric matrix B. Let 0 < µ1 ≤ . . . ≤ µ4 be the eigenvalues of B, and write k.k for the usual Euclidean 2-norm. If Q(x) = `, then kAxk = kB T Bxk  µ4 `1/2 = (λ4 `)1/2  (∆`)1/2 and kxk  (`/λ1 )1/2  `1/2 by (3.5) and (3.6). Let V = (vij ) ∈ R2×4 , say, be the matrix whose two rows are given by vT1 A and vT2 A. It is not hard to see that (3.10) implies   2 X det v1j1 v1j2 ≥ µ1 µ2  1. v2j1 v2j2 1≤j1